Is it obligatory to stop working as soon as one doubts its unlawfulness, or is it permissible to continue working until its unlawfulness is proven, and what must be done after its unlawfulness is proven?

The ruling on this depends on the reason for the doubt. If there is a legitimate reason for the doubt and you are strongly inclined to believe it is forbidden, then you must stop working until its permissibility becomes clear to you. If there is no legitimate reason for the doubt, then it should be disregarded. If it is proven to be forbidden after you have engaged in it, then you must cease from it, and what you earned from it before knowing its prohibition is permissible for you to benefit from.